Definition:Ideal (Ring Theory)

From ProofWiki

Jump to: navigation, search

Definition

Let $\left({R, +, \circ}\right)$ be a ring, and let $\left({J, +}\right)$ be a subgroup of $\left({R, +}\right)$.

Then $J$ is an ideal of $R$ iff:

$\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$

The letter $J$ is frequently used to denote an ideal.

$J$ is a left ideal of $R$ iff:

$\forall j \in J: \forall r \in R: r \circ j \in J$

$J$ is a right ideal of $R$ iff:

$\forall j \in J: \forall r \in R: j \circ r \in J$

It follows that in a commutative ring, a left ideal, a right ideal and an ideal are the same thing.

A proper ideal $J$ of $\left({R, +, \circ}\right)$ is an ideal of $R$ such that $J$ is a proper subset of $R$.

That is, such that $J \subseteq R$ and $J \ne R$.